Humanitext Reader

Euclid · Elements §6.prop.5

Equiangularity of Triangles with Proportional Sides

Passage 88 of 316 · Greek

Summary

Proves that if two triangles have their sides proportional, they are equiangular and their corresponding angles are equal (the SSS similarity theorem for triangles).

§6.prop.5ἐὰν δύο τρίγωνα τὰς πλευρὰς ἀνάλογον ἔχῃ, ἰσογώνια ἔσται τὰ τρίγωνα καὶ ἴσας ἕξει τὰς γωνίας, ὑφʼ ἃς αἱ ὁμόλογοι πλευραὶ ὑποτείνουσιν.
If two triangles have their sides proportional, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend.
ἔστω δύο τρίγωνα τὰ ΑΒΓ, ΔΕΖ τὰς πλευρὰς ἀνάλογον ἔχοντα, ὡς μὲν τὴν ΑΒ πρὸς τὴν ΒΓ, οὕτως τὴν ΔΕ πρὸς τὴν ΕΖ, ὡς δὲ τὴν ΒΓ πρὸς τὴν ΓΑ, οὕτως τὴν ΕΖ πρὸς τὴν ΖΔ, καὶ ἔτι ὡς τὴν ΒΑ πρὸς τὴν ΑΓ, οὕτως τὴν ΕΔ πρὸς τὴν ΔΖ. λέγω, ὅτι ἰσογώνιόν ἐστι τὸ ΑΒΓ τρίγωνον τῷ ΔΕΖ τριγώνῳ καὶ ἴσας ἕξουσι τὰς γωνίας, ὑφʼ ἃς αἱ ὁμόλογοι πλευραὶ ὑποτείνουσιν, τὴν μὲν ὑπὸ ΑΒΓ τῇ ὑπὸ ΔΕΖ, τὴν δὲ ὑπὸ ΒΓΑ τῇ ὑπὸ ΕΖΔ καὶ ἔτι τὴν ὑπὸ ΒΑΓ τῇ ὑπὸ ΕΔΖ. συνεστάτω γὰρ πρὸς τῇ ΕΖ εὐθείᾳ καὶ τοῖς πρὸς αὐτῇ σημείοις τοῖς Ε, Ζ τῇ μὲν ὑπὸ ΑΒΓ γωνίᾳ ἴση ἡ ὑπὸ ΖΕΗ, τῇ δὲ ὑπὸ ΑΓΒ ἴση ἡ ὑπὸ ΕΖΗ· λοιπὴ ἄρα ἡ πρὸς τῷ Α λοιπῇ τῇ πρὸς τῷ Η ἐστιν ἴση.
Let ABC, DEF be two triangles having their sides proportional, so that, as AB is to BC, so is DE to EF, as BC is to CA, so is EF to FD, and further, as BA is to AC, so is ED to DF. I say that the triangle ABC is equiangular to the triangle DEF and they will have those angles equal which the corresponding sides subtend, namely the angle ABC to the angle DEF, the angle BCA to the angle EFD, and further the angle BAC to the angle EDF. For let there be constructed on the straight line EF and at the points E, F on it, the angle FEG equal to the angle ABC, and the angle EFG equal to the angle ACB; therefore the remaining angle at A is equal to the remaining angle at G.
ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΓ τρίγωνον τῷ ΕΗΖ.
Therefore the triangle ABC is equiangular to the triangle EGF.
τῶν ἄρα ΑΒΓ, ΕΗΖ τριγώνων ἀνάλογόν εἰσιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας καὶ ὁμόλογοι αἱ ὑπὸ τὰς ἴσας γωνίας ὑποτείνουσαι· ἔστιν ἄρα ὡς ἡ ΑΒ πρὸς τὴν ΒΓ, ἡ ΗΕ πρὸς τὴν ΕΖ. ἀλλʼ ὡς ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως ὑπόκειται ἡ ΔΕ πρὸς τὴν ΕΖ·
Therefore in the triangles ABC, EGF the sides about the equal angles are proportional, and those subtending the equal angles are corresponding; therefore, as AB is to BC, so is GE to EF.
ὡς ἄρα ἡ ΔΕ πρὸς τὴν ΕΖ, οὕτως ἡ ΗΕ πρὸς τὴν ΕΖ. ἑκατέρα ἄρα τῶν ΔΕ, ΗΕ πρὸς τὴν ΕΖ τὸν αὐτὸν ἔχει λόγον·
But, as AB is to BC, so by hypothesis is DE to EF; therefore, as DE is to EF, so is GE to EF.
ἴση ἄρα ἐστὶν ἡ ΔΕ τῇ ΗΕ. διὰ τὰ αὐτὰ δὴ καὶ ἡ ΔΖ τῇ ΗΖ ἐστιν ἴση.
Therefore each of the straight lines DE, GE has the same ratio to EF; therefore DE is equal to GE. For the same reason, DF is also equal to GF.
ἐπεὶ οὖν ἴση ἐστὶν ἡ ΔΕ τῇ ΕΗ, κοινὴ δὲ ἡ ΕΖ, δύο δὴ αἱ ΔΕ, ΕΖ δυσὶ ταῖς ΗΕ, ΕΖ ἴσαι εἰσίν· καὶ βάσις ἡ ΔΖ βάσει τῇ ΖΗ ἴση· γωνία ἄρα ἡ ὑπὸ ΔΕΖ γωνίᾳ τῇ ὑπὸ ΗΕΖ ἐστιν ἴση, καὶ τὸ ΔΕΖ τρίγωνον τῷ ΗΕΖ τριγώνῳ ἴσον, καὶ αἱ λοιπαὶ γωνίαι ταῖς λοιπαῖς γωνίαις ἴσαι, ὑφʼ ἃς αἱ ἴσαι πλευραὶ ὑποτείνουσιν.
Since then DE is equal to EG, and EF is common, therefore the two sides DE, EF are equal to the two sides GE, EF respectively; and the base DF is equal to the base FG; therefore the angle DEF is equal to the angle GEF, and the triangle DEF is equal to the triangle GEF, and the remaining angles are equal to the remaining angles, which the equal sides subtend.
ἴση ἄρα ἐστὶ καὶ ἡ μὲν ὑπὸ ΔΖΕ γωνία τῇ ὑπὸ ΗΖΕ, ἡ δὲ ὑπὸ ΕΔΖ τῇ ὑπὸ ΕΗΖ. καὶ ἐπεὶ ἡ μὲν ὑπὸ ΖΕΔ τῇ ὑπὸ ΗΕΖ ἐστιν ἴση, ἀλλʼ ἡ ὑπὸ ΗΕΖ τῇ ὑπὸ ΑΒΓ, καὶ ἡ ὑπὸ ΑΒΓ ἄρα γωνία τῇ ὑπὸ ΔΕΖ ἐστιν ἴση.
Therefore the angle DFE is also equal to the angle GFE, and the angle EDF to the angle EGF. And since the angle FED is equal to the angle GEF, while the angle GEF is equal to the angle ABC, therefore the angle ABC is also equal to the angle DEF.
διὰ τὰ αὐτὰ δὴ καὶ ἡ ὑπὸ ΑΓΒ τῇ ὑπὸ ΔΖΕ ἐστιν ἴση, καὶ ἔτι ἡ πρὸς τῷ Α τῇ πρὸς τῷ Δ· ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΓ τρίγωνον τῷ ΔΕΖ τριγώνῳ.
For the same reason, the angle ACB is also equal to the angle DFE, and further the angle at A to the angle at D; therefore the triangle ABC is equiangular to the triangle DEF.
ἐὰν ἄρα δύο τρίγωνα τὰς πλευρὰς ἀνάλογον ἔχῃ, ἰσογώνια ἔσται τὰ τρίγωνα καὶ ἴσας ἕξει τὰς γωνίας, ὑφʼ ἃς αἱ ὁμόλογοι πλευραὶ ὑποτείνουσιν· ὅπερ ἔδει δεῖξαι.
Therefore, if two triangles have their sides proportional, the triangles will be equiangular and will have those angles equal which the corresponding sides subtend; which was to be proved.

Notes

  1. §6.prop.5ὑφʼ ἃς — The preposition ὑπό governing the accusative relative pronoun ἅς (feminine plural, referring back to the preceding γωνίας). The subject of the relative clause is αἱ ὁμόλογοι πλευραὶ (the corresponding sides). It is a standard mathematical idiom meaning "subtended by (the sides)."
  2. §6.prop.5συνεστάτω — Third-person singular present passive imperative of συνίστημι (to construct). Although followed by two subjects (ἡ ὑπὸ ΖΕΗ and ἡ ὑπὸ ΕΖΗ), the verb is in the singular form, which is a common construction in Greek mathematics when introducing several elements. It translates as "let there be constructed..."
  3. §6.prop.5λοιπὴ ἄρα ἡ πρὸς τῷ Α λοιπῇ τῇ πρὸς τῷ Η ἐστιν ἴση. — The subject is the nominative λοιπὴ ἡ πρὸς τῷ Α (the remaining angle at A), which is equal to the dative λοιπῇ τῇ πρὸς τῷ Η (to the remaining angle at H). This relies on the geometric axiom that if two angles of one triangle are equal to two angles of another, their remaining third angles must also be equal.
  4. §6.prop.5ἑκατέρα ἄρα τῶν ΔΕ, ΗΕ πρὸς τὴν ΕΖ τὸν αὐτὸν ἔχει λόγον — The nominative singular feminine ἑκατέρα (each) is the subject, governing the genitive τῶν ΔΕ, ΗΕ. It states that each of the two magnitudes (DE and HE) has the same ratio to the third (EZ), which dynamically leads to the conclusion that DE must be equal to HE.

Cite this passage

Euclid, Elements §6.prop.5. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:6.prop.5

Please note the AI-draft status of the translation and the date accessed.

Translation, notes and summary are AI-generated drafts, revised through reader feedback.